The Timing Capacity of Single-Server Queues with Multiple Flows

Transcription

1 The Timing Capacity of Single-Server Queues with Multiple Flows Xin Liu and R. Srikant Coordinated Science Laboratory University of Illinois at Urbana Champaign March 14, 2003

2 Timing Channel Information can be transmitted through the timing-intervals between messages/events

3 Distortion Distortion of timing information Queueing is a mechanism that naturally blurs the timing information 2

4 Multiple Flows What is the sum timing capacity? 3

5 Interference Flow What is the timing capacity of a flow when there exists uncontrollable and undetectable cross traffic? 4

6 An Exponential Server Queue Interference flow: Poisson arrival with rate λ I Service time distribution for all packets: i.i.d. exponentially distributed with mean 1/µ D1 D 2 D 3 A 1 A 2 A 3 5

7 A Lower Bound on Capacity Service discipline: FIFO A lower bound on the timing capacity is ( ) µ λi C L (λ 0 ) = λ 0 log, λ 0 where λ 0 + λ I µ. Input process: Poisson with rate λ 0. Special case: λ I = 0 C(λ 0 ) = λ 0 log µ. λ 0 6

8 Intuition D1 D 2 D 3 S 1 S 2 S 3 A 1 A 2 A 3 Randomness is caused by queue and service time Effective service time is exponentially distributed with mean 1/(µ λ I ). 7

9 Proof I(A n ; D n ) = h(d n ) + h(a n ) h(d n, A n ) (1) = h(d n ) + h(a n ) h(s n, A n ) = h(d n ) h(s n A n ) h(d n ) h(s n ) n h(d n ) h(s i ) (2) = = n i=1 n i=1 i=1 ( log 1 ) + 1 λ 0 log µ λ I λ 0, 8 n i=1 ( log ) µ λ I

10 Number of Effective Interfering Packets n i : number of effective interfering packets P (n i = k) = π(j)p(k j) = p(k k)π(k) + j=k j=k+1 = (1 ρ)ρ k (1 q 0 ) k + = ( λi µ ) k ( 1 λ I µ p(k j)π(k) j=k+1 ), k = 0, 1, 2, (1 ρ)ρ j (1 q 0 ) k q 0 q 0 = λ 0 λ 0 +λ I : probability a packet belongs to flow 0 9

11 Effective Service Time n i + 1: geometrically distributed with mean µ/(µ λ I ) Effective service time: sum of n i + 1 independent and exponentially distributed random variable is exponential with mean E(S i ) = 1 µ E(n i + 1) = 1 µ λ I. 10

12 Multiple Flows N: number of flows B = log N bits for address Service times are i.i.d. exponentially distributed. 11

13 A Lower Bound The arrival process of each flow is an independent Poisson process with rate λ i, λ i µ. Consider all other flows as interference. 12

14 A Lower Bound Cont d We have C i λ i log ( µ j i λ ) j. λ i Lower bound is maximized when all users have the same arrival rate. Maximize over λ C (B 1 log B)µ. 13

15 Theorem Theorem: The timing capacity of the N flows satisfies (B 1 log B)µ C Bµ. Upper bound holds because the overall information capacity cannot exceed Bµ for B 2 bits. 14

16 The Upper Bound X n : information sent through the packets I(X n, D n ; X n, A n ) = I(D n ; X n, A n ) + I(X n ; X n, A n D n ) (1) = I(D n ; A n ) + I(X n ; X n ) (2) µb, (1): X n contains no additional information regarding D n other than that in A n. (2): if B > 1 bit, the system capacity is µb. 15

17 Timing Capacity of Multiple Flows The arrival process of each flow is an independent Poisson process with rate λ, Nλ µ. The lower bound is asymptotically tight. Timing capacity increases as the number of flows increases. 16

18 A Single Flow Each packet has B bits All B bits are used to distinguish sub-flows; i.e. there are N = 2 B sub-flows 17

19 Timing Capacity of A Single Flow We have (B 1 log B)µ C T Bµ. The timing capacity is close to the server capacity Bµ bits/sec Without splitting, it is µ bits/sec A large amount of information can be conveyed through timing. When λ is small, the distortion caused by queueing delay is relatively small. 18

20 Covert Information Eavesdropper monitors the server, records packets in sequence 19

21 Covert Information Covert information C c : C c = C T C E, C T : information rate at the receiver C E : information rate at the eavesdropper Covert information: secrets that cannot be heard by the eavesdropper. 20

22 Two Flows I(A n, B m ; N n+m, D n+m ) = h(a n, B m ) + h(n n+m, D n+m ) h(a n, B m, N n+m, D n+m ) h(a n, B m ) + h(n n+m, D n+m ) h(a n, B m, D n+m ) = h(a n, B m ) + h(n n+m, D n+m ) h(a n, B m, S n+m ) = h(n n+m, D n+m ) h(s n+m ) h(d n+m ) h(s n+m ) + H(N n+m ). 21

23 Covert Information Cont d I(A n, B m ; N n+m ) = H(N n+m ) FIFO Eavesdropper located at the input of server. Covert information C c = C T C E λ 1 + λ 2 ( h(d n+m ) h(s n+m ) ), n + m which is the covert information of a single flow with rate λ 1 + λ 2. 22

24 Location of the Eavesdropper 23

25 A Special Case 24

26 Service Disciplines First come first serve: covert information rate cannot be larger than that of a single flow. Random service discipline: covert information rate is larger than that of a single flow. Intuition: timing information reduces randomness introduced by the service discipline. Implementation: each packet randomly picks a diffserv class in its header. 25

27 Service Disciplines Cont d I(A n, B m ; N n+m, D n+m ) = h(a n, B m ) + h(n n+m, D n+m ) h(a n, B m, N n+m, D n+m ) = h(a n, B m ) + h(n n+m, D n+m ) h(a n, B m, S n+m, N n+m ) = h(a n, B m ) + h(n n+m, D n+m ) h(a n, B m, S n+m ) = h(n n+m, D n+m ) h(s n+m ) = h(d n+m ) h(s n+m ) + h(n n+m ). 26

28 Random Service Discipline Total information: I(A n, B m ; N n+m, D n+m ) = h(d n+m ) h(s n+m ) + h(n n+m ) h(n n+m A n, B m, S n+m ) Eavesdropper: I(N n+m ; A n, B m ) = h(n n+m ) h(n n+m A n, B m ). Covert information: h(d n+m ) h(s n+m )+h(n n+m A n, B m ) h(n n+m A n, B m, S n+m ). 27

29 Random Service Discipline Cont d h(d n+m ) h(s n+m ) is maximized when the input is Poisson. h(n n+m A n, B m ) h(n n+m A n, B m, S n+m ) is positive because N n+m is not independent of S n+m conditioned on (A n, B m ). 28

30 Discrete-Time Case N = 2 B : number of flows Geometric service time with mean 1/µ: µ(b 1) µ log(b 1) C µb + 1 Deterministic service time (one packet/slot): (B 1) log(b 1) C B

31 General Case General service-time distribution: ( B log B 1 B µ ( 1 + Bµ2 E(S 2 ) 2 )) where E(S 2 ) is the second moment of the service-time C Bµ + 1, Queueing statistics of a general server queue is unknown. Basic idea: use waiting time + service time as an upper bound for the effective service time. Good approximation for small λ. 30

32 Conclusion An asymptotically tight lower bound on the timing capacity in the presence of interference traffic Timing information for multiple flows Continuous case Discrete case Coloring increases the timing information conveyed by a single flow. The location of the eavesdropper is important. It can significantly decrease the amount of covert information. 31

33 Note h(n n+m A n, B m ) h(n n+m A n, B m, S n+m ) = n+m i=1 I(N i ; D n+m A n, B m, N i 1 ). I(N i ; D n+m A n, B m, N i 1 ) = n+m j=1 I(N i ; D j A n, B m, D j 1, N i 1 ) I(N i ; D i 1 A n, B m, D i 2, N i 1 ) ( = f(a n, B m, D i 2 )E log p(n i D i 1, A n, B m, D i 2 ) ) p(n i A n, B m, D i 2. ) To show I(N i ; D n+m A n, B m, N i 1 ) is positive, we only need to show that with a positive probability p(n i = m D i 1, A n, B m, D i 2, N i 1 ) p(n i = m A n, B m, D i 2, N i 1 ), m = 1, 2. Consider the case where after the departure of (i 2)th packet, there is more than 1 packet in the queue. This event happens with a positive probability. Consider two events with positive probabilities: 1) during the service time of the 32

34 (i 1)th packet, no new packet arrives. 2) during the service time of the (i 1)th packet, one new packet arrives. Apparently, in these two cases, p(n i = m D i 1, A n, B m, D i 2, N i 1 ) is different. Thus, p(n i = m D i 1, A n, B m, D i 2, N i 1 ) p(n i = m A n, B m, D i 2, N i 1 ) with a positive probability. 33